The derived category \(\mathcal {C}\)  admits a large group of auto-equivalences \(\operatorname{Aut}(\mathcal {C})\) , generated by the
homological shift \(E\mapsto E[1]\) , by the translation \(E\mapsto E(1)=E \otimes \mathcal {O}_X(1)\) ,
by spherical twists \(ST_\mathcal {O}\)  with respect to the spherical object \(\mathcal {O}\) ,
and by automorphisms of \(X\) 
itself (or rather, its formal completion at \({P}^2\) ). For any spherical object \(S\) , the spherical twist
\(\operatorname{ST}_S\)  acts on \(E\in \mathcal {C}\)  via [1]}
\(\operatorname{ST}_S\colon E \mapsto \operatorname{Cone}\left(\operatorname{Hom}^{\bullet }_{K_{{P}^2}}(S,E) \otimes S \stackrel{\rm ev}{\rightarrow } E \right)\) 
